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A335384
Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.
0
6, 48, 168, 180, 480, 2016, 3528, 5760, 11232, 13200, 20160, 26208, 61200, 78336, 123120, 181440, 267168, 374400, 511056, 682080, 892800, 1014816, 1488000, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 9999360, 11908560, 13615200, 16511040, 19845936, 24261120, 25048800, 28003968
OFFSET
1,1
COMMENTS
GL(m,q) is the general linear group, the group of invertible m X m matrices over the finite field F_q with q = p^k elements.
By definition, all fields must contain at least two distinct elements, so q >= 2. As GL(1,q) is isomorphic to F_q*, the multiplicative group [whose order is p^k-1 (A181062)] of finite field F_q, data begins with m >= 2.
Some isomorphisms (let "==" denote "isomorphic to"):
a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.
a(2) = 48 for GL(2,3) that has 55 subgroups.
a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].
a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).
Array for order of GL(m,q) begins:
=============================================================
m\q | 2 3 4=2^2 5 7
-------------------------------------------------------------
2 | 6 48 180 480 2016
3 | 168 11232 181440 1488000 33784128
4 | 20160 24261120 2961100800 116064000000 #GL(4,7)
5 |9999360 #GL(5,3) ... ... ...
REFERENCES
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.
FORMULA
#GL(m,q) = Product_{k=0..m-1}(q^m-q^k).
EXAMPLE
a(1) = #GL(2,2) = (2^2-1)*(2^2-2) = 3*2 = 6 and the 6 elements of GL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 invertible matrices with entries in F_2:
(1 0) (1 1) (1 0) (0 1) (0 1) (1 1)
(0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.
CROSSREFS
Cf. A059238 [GL(2,q)].
Cf. A002884 [GL(m,2)], A053290 [GL(m,3)], A053291 [GL(m,4)], A053292 [GL(m,5)], A053293 [GL(m,7)], A052496 [GL(m,8)], A052497 [GL(m,9)], A052498 [GL(m,11)].
Cf. A316622 [GL(n,Z_k)].
Sequence in context: A250274 A167547 A244726 * A331668 A005353 A047927
KEYWORD
nonn
AUTHOR
Bernard Schott, Jun 04 2020
STATUS
approved